The discovery of the integer quantum Hall effect in the early eighties of the last century, with highly precise quantization values for the Hall conductance in multiples of e²/h, has been the first fascinating manifestation of the topological state of matter driven by magnetic field and disorder, and related to the formation of non-dissipative current flow. Throughout the 2000’s, several new phenomena such as the spin Hall effect and the quantum spin Hall effect were confirmed experimentally for systems with strong spin-orbit coupling effects and in the absence of external magnetic field. More recently, the Zeeman spin Hall effect and the formation of valley Hall topological currents have been introduced for graphene-based systems, under time-reversal or inversion symmetry-breaking conditions, respectively. This review presents a comprehensive coverage of all these Hall effects in disordered graphene from the perspective of numerical simulations of quantum transport in two-dimensional bulk systems (by means of the Kubo formalism) and multiterminal nanostructures (by means of the Landauer-Büttiker scattering and non-equilibrium Green’s function approaches). In contrast to usual two-dimensional electron gases in semiconductor heterostructures, the presence of defects in graphene generates more complex electronic features such as electron-hole asymmetry, defect-induced resonances in the electron density of states or percolation effect between localized impurity states, which, together with extra degrees of freedom (sublattice pseudospin and valley isospin), bring a higher degree of complexity and enlarge the transport phase diagram.

中文翻译：

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无序石墨烯中的电荷，自旋和谷值霍尔效应

上世纪八十年代初发现整数量子霍尔效应，并以e ² / h的倍数对霍尔电导进行了高度精确的量化，是磁场和无序驱动的物质拓扑状态的第一个引人入胜的表现，并且与非耗散电流的形成有关。在整个2000年代，对于具有强自旋轨道耦合效应且没有外部磁场的系统，实验上证实了一些新现象，如自旋霍尔效应和量子自旋霍尔效应。最近，已经针对基于石墨烯的系统引入了Zeeman自旋霍尔效应和谷底霍尔拓扑电流的形成，分别处于时间反转或反转对称破坏条件下。这篇综述从二维本体系统（通过Kubo形式主义）和多末端纳米结构（通过Landauer-Büttiker散射和扩散）的量子输运的数值模拟的角度，全面介绍了无序石墨烯中的所有这些霍尔效应。非平衡格林函数法）。与半导体异质结构中通常的二维电子气相反，石墨烯中缺陷的存在会产生更复杂的电子特征，例如电子-空穴不对称性，缺陷在电子态的密度中引起的共振或局部杂质态之间的渗流效应，这加上额外的自由度（亚晶格伪自旋和谷间等旋）带来了更高的复杂度并扩大了运输阶段图。